Students integrate numerically to find the electric field due to a cone of surface charge, and then visualize the result. This integral can be done in either spherical or cylindrical coordinates, giving students a chance to reason about which coordinate system would be more convenient.

Students calculate the flux from the vector field \(\vec{F} = C\, z\, \hat{z}\) through a right cone of height \(H\) and radius \(R\) .

Students set up and compute a scalar surface integral.

1. A right circular cone has circular base of radius R and height H, both measured in feet.
   1. What is the volume of the cone?
   2. Write down as many different integrals as you can for computing this volume.
   3. Do at least two of these integrals.

For some integrals, you may wish to use the fact that \[\cos(2\alpha)=2\cos^2\!\alpha-1=1-2\sin^2\!\alpha\]

None